Rothwell then spent last season on loan at Leeds United, once again winning promotion out of the Championship and making 36 appearances along the way.
With one year left on his deal, Gers chiefs are in talks with Bournemouth over a permanent deal as reported by Rangers Review.
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Rothwell would sign a two-year deal at Ibrox if the discussions are successful.
The Light Blues have already signed one player from Bournemouth in this window, right-back Max Aarons on a year-long loan deal. He is another known to Martin after playing together at Norwich.
Lyall Cameron is the other summer signing to date and would be in direct competition with Rothwell in the middle of the park, along with Connor Barron, Nico Raskin, Mohamed Diomande, Nedim Bajrami and Bailey Rice.
Meanwhile, Rangers are close to appointing ex-Arsenal goalkeeping coach Sal Bibbo as part of Russell Martin’s staff.